NONEQUALITY OF DIMENSIONS FOR METRIC SPACES BY PRABIR ROY(l) Introduction. There are three classical set-theoretic notions of dimension; they are [2, p. 153]: Small inductive dimension ( = Menger-Urysohn dimension), denoted by ind such that ind (S) = —1 if S is empty, ind (S) = n if for every point p e S and open set U containing p there is an open set V satisfying p e V <=■U, ind (boundary of V) S n-l, and ind (S)=n if ind (S)fZn but ind (S) = n- 1 is not true. Large inductive dimension (due to Urysohn), denoted by Ind such that Ind (S) = - 1 if S is empty, Ind (S) á n if for every closed set Ce S and open set U con- taining C there is an open set V satisfying Cc v= U, Ind (boundary of V) = n-1, and Ind (S) = n if Ind (S)^n but Ind (S)Sn-l is not true. Covering dimension (=Lebesgue covering dimension), denoted by dim such that dim (S) = - 1 if S is empty, dim (S) S n if every finite open cover of S has a finite open refining cover of order ^n+1, that is, no point of S belongs to more than n -I-1 members of the refinement, and dim (S)=n if dim (S) ^ n but dim (S) ^ n -1 is not true. It is well known that for separable metric space S ind (S) = Ind (S) = dim (S). Recently Katetov showed that for any metric space S Ind (S) = dim (S). However, the question, is ind (S)=dim (S) for all metric spaces, remained open. We shall answer this question in the negative. (For a comprehensive account of the preceding material the reader is invited to consult P. Aleksandrov's paper [1, pp. 1-4].) We shall prove, in succeeding sections, the following statement. Theorem. There is a complete metric space A such that ind (A)=0 but dim (A) = 1. 1. Description of the space. In this section we define the points of A and certain subsets of A. Although these subsets are already called regions here, the fact that Presented to the Society, January 26, 1963 under the title Failure of equivalence of dimension concepts for metric spaces; received by the editors April 17, 1967. (') Supported by National Science Foundation grants NSF-G21514 and NSF-GP-6951. 117 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 118 PRABIR ROY [October they form a topological basis is proved in §2. The remarks at the end are intended to clarify some of the rather arbitrary assertions implicit in the process of defining. 1.1. Notations. The symbol 0 as usual will denote the empty set. By a sequence we shall mean a function defined on either the set of all positive integers, or the set of all nonnegative integers, or any initial segment of either of them. With this in mind the following notations are adopted. 1.1.1. A"=the set of all finite sequences of real numbers, defined on initial seg- ments of the nonnegative integers, such that if xe X, then x(i)=0 only in case /=0. 1.1.2. If xe A', then \x\ will denote the greatest integer for which x is defined. (I*! may be thought of as the length of x.) 1.1.3. y=the set of all reversible sequences of positive numbers, defined on the set of all positive integers. The word "reversible" will be used instead of "one- to-one." 1.1.4. Z=the set of all infinite sequences of positive numbers, defined on the set of all positive integers. 1.1.5. If r is a positive number, then Yr will denote the set of all members of Y which take on the value r, and 1.1.6. FT will denote a reversible function from the positive numbers onto Yr. 1.1.7. If A" is a collection of sets, then K* will denote the union of members' of a:. 1.2. Points of A. There are two types of points of A and these types will be called Px and P2. 1.2.1. Px = the set of all sequences of nonzero real numbers, defined on the posi- tive integers. 1.2.2. P2 = Xx YxZ. (px,Pv, and pz will denote the coordinates of peP2.) 1.3. Regions of A. There are also two types of regions and these types will be denoted by Tx and T2. Risa member of Tx only in case there is an x e X with |x| #0 such that, 1.3.1. R = R1kj R2, where 1.3.2. R1 = {p \pePx; and p(i) = x(i) for z'=l,..., \x\}, and 1.3.3. R2={p\peP2; \px\^\x\; and px(i)=x(i) for i=l,..., \x\}. Such a region R will be denoted by Rx. Ris a member of T2 only in case there is a point p eP2 and a positive integer n such that, 1.3.4. R = R° u R* u R-, where 1.3.5. R° = {q\qeP2; qx=Px\ Çy=Py; and if n>\, then qz(i)=pz(i) for i=l,..., n-\}, and 1.3.6. R* =(Jf=i **,.«. +>, and R~ = U" i A*,.».-),, where {y(p,n, +),}f and {y(p, n, -\)x are the two infinite reversible sequences of members of A'such that if y is a positive integer, then 1.3.7. \y(p,n, ±),\ = \px\+n+l, License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 1968] NONEQUALITY OF DIMENSIONS FOR METRIC SPACES 119 1.3.8. y(p, n, ±)j(i)=Px(i) for ¿=0,..., \px\, 1.3.9. y(p,n, ±)f(\px\ + l)= ±pA\n+j-\), 1.3.10. y(p,n, ±)f(\Px\+2)= + Fp-Y\n+f-»(pY),andifn>l, 1.3.11. y(p,n, ±)f(\px\+2 + i)=+pz(i) for i=l,...,n-l. Such a region 7? will be denoted by 7?(p>n). 1.4. Remarks. Note that 1.1.6 makes sense because Yr has the power of the continuum. In 1.2, Px n F2 = 0. In 1.3, the fact that {y(p, n, + )¡}f and {y(p, n, - )(}f are reversible sequences follows from 1.3.9, 1.1.3, and 1.1.1, whereby pY is a re- versible sequence of positive numbers, and two members of X are different if they differ for some integer. Also for the same reason we have that y(p, n, +), / y(p, n, —)kfor each of y and k an integer. Finally observe that item 1.3.10 makes sense because by 1.1.5 pYe YPYin+j-X)and by 1.1.6, FPy(n+ j.X) is a reversible function onto YPr(n+f_X). 2. Some preliminary lemmas. Now we shall exhibit, in a series of lemmas, certain basic properties of the space A, including a proof that the regions defined in §1, are indeed well defined. In subsequent sections these lemmas will be translated into statements asserted in the theorem. 2.1. Lemma. Suppose that each of Rx and Rx. is a member of Tx and each of R(p¡n)and 7?(,>m)is a member of T2. 2.1.1. RX=>RX oRxr\ Rx.^0 and \x\ ^ |x'| -=- |x| = \x'\ and x(i)=x'(i) for /=!,..., |x|. 2.1.2. Rx=>Rip,n) <>Rxn 7?(°p>n)#0 o Rx^RPx. 2.1.3. 7?(pn)^>RX o Rtp.n)=>Px- 2.1.4. 7î(P,n)n7î(,,m)#0, \px\ = \qx\, and n^mon = m and R°ip_n)n 7%,m) ¥=0 o R(p.n)=>R(g,m)and \px\ = \qx\. 2.1.5. R(p_n)=>RiQim)and \px\ <\qx\ o RtP,n)=>R(q,my Proof. In each item we shall prove the implications pointing to the right and finally show that the last statement implies the first one. Proof of 2.1.1. For the first implication, by 1.1.3, 1.2.2, and 1.1.1, let p e Rx. with p e P2 andpx = x'. Since peRx, by 1.3.3, we have |x'| = |/>x| = \x\. The next implication is immediate by applying 1.3.2 if a point p ePx belongs to the inter- section, and by applying 1.3.3 if a point/» £ P2 belongs to the intersection. That the first statement follows from the last one is obvious by the definition of Tx. Proof of 2.1.2. The first implication is transparent. For the second implication let />' belong to the common part. Noting that p'x=px by 1.3.5 and |/?x|^|x| by 1.3.3, we have that p' e RPx by 1.3.3 and hence Rx=>RPxby 2.1.1. In the last im- plication observe that RPx=>R°PMby 1.3.5 and 1.3.3; and in view of 2.1.1, RPx =>7?á,n>by1.3.6, 1.3.7,and 1.3.8. Proof of 2.1.3. Assume that the first implication is false. By 1.1.1 and 1.2.2, let p' eP2 with p'x=px and p'y¥=Py-It follows that p' eRPx, by 1.3.3; and hence License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 120 PRABIR ROY [October p' e Rxby 2.1.1, in view of the assumption, but/)' $ F(p-n),by 1.3.5, 1.3.7 and 1.3.3.